In general, for any two objects in space, a given object 1 radiates
to object 2, and to other places as well, as shown in
Figure 19.10.

Figure 19.10:
Radiation between two bodies

Figure 19.11:
Radiation between two
arbitrary surfaces

We want a general expression for energy interchange between two
surfaces at different temperatures. This is given by the radiation
shape factor or view factor,
. For the
situation in Figure 19.11,

=

fraction of energy leaving 1 which
reaches 2

=

fraction of energy leaving 2 which reaches 1

,

are
functions of geometry only

For body 1, we know that
is the emissive power of a black
body, so the energy leaving body 1 is
. The energy
leaving body 1 and arriving (and being absorbed) at body 2 is
. The energy leaving body 2 and being absorbed
at body 1 is
. The net energy interchange from
body 1 to body 2 is

(19..4)

Suppose both surfaces are at the same temperature so there is no net
heat exchange. If so,

but also
. Thus

Equation (19.4) is the
shape factor reciprocity relation. The net heat exchange
between the two surfaces is

We know that
, i.e., that all of the energy emitted by
1 gets to 2. Thus

This can be used to find the net heat transfer from 2 to 1.

View factors for other configurations can
be found analytically or numerically. Shape factors are given in
textbooks and reports (they are tabulated somewhat like Laplace
transforms), and examples of the analytical forms and numerical
values of shape factors for some basic engineering configurations
are given in Figures 19.13 through
19.16, taken from the book by
Incropera and DeWitt.

Figure:
Total emittances for
different surfaces [from: A Heat Transfer Textbook, J.
Lienhard]